Subgroups — C₃×C₃⋊(C₃⋊Dic₁₅) #13108508

Order 1620.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⁴×C₅	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊Dic₁₅)	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₃⋊Dic₁₅	—	3	13	no	maximal
4	C₃⁴×C₅	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×C₃⋊(C₃⋊Dic₃)	—	5	1	no	maximal, Hall
6	C₂×C₃³×C₅	—	2	13	no	—
7	C₂×C₃³×C₅	D₃	1	13	yes	—
8	C₂×C₃³×C₅	C₂×C₃	1	1	yes	characteristic
9	C₃⋊Dic₁₅	—	3	13	no	—
10	C₃×Dic₁₅	—	9	13	no	—
11	C₂×C₃⁴	D₅	1	1	yes	characteristic
12	C₃³×C₅	—	2	13	no	—
13	C₃³×C₅	Dic₃	1	13	yes	complemented
14	C₃³×C₅	C₃×C₄	1	1	yes	characteristic, complemented
15	C₃⋊(C₃⋊Dic₃)	—	5	1	no	—
16	C₃×C₃⋊Dic₃	—	15	13	no	—
17	C₂×C₃²×C₅	—	2	52	no	—
18	C₂×C₃²×C₅	C₃×D₃	1	13	yes	—
19	C₂×C₃²×C₅	C₃⋊D₃	1	13	yes	—
20	C₃⁴	Dic₅	1	1	yes	characteristic, complemented, Sylow
21	Dic₁₅	—	9	13	no	—
22	C₃×Dic₅	—	27	1	no	—
23	C₂×C₃³	—	2	13	no	—
24	C₂×C₃³	C₃×D₅	1	1	yes	characteristic
25	C₂×C₃³	D₁₅	1	13	yes	—
26	C₃²×C₅	—	2	52	no	—
27	C₃²×C₅	C₃×Dic₃	1	13	yes	complemented
28	C₃²×C₅	C₃⋊Dic₃	1	13	yes	complemented
29	C₃⋊Dic₃	—	15	13	no	—
30	C₃×Dic₃	—	45	13	no	—
31	C₂×C₃×C₅	—	2	13	no	—
32	C₂×C₃×C₅	C₃×C₃⋊D₃	1	13	yes	—
33	C₂×C₃×C₅	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
34	C₃³	—	2	13	no	—
35	C₃³	C₃×Dic₅	1	1	yes	characteristic, complemented
36	C₃³	Dic₁₅	1	13	yes	complemented
37	Dic₅	—	27	1	no	Hall
38	C₂×C₃²	—	2	52	no	—
39	C₂×C₃²	C₃×D₁₅	1	13	yes	—
40	C₂×C₃²	C₃⋊D₁₅	1	13	yes	—
41	C₃×C₅	—	2	13	no	—
42	C₃×C₅	C₃×C₃⋊Dic₃	1	13	yes	complemented
43	C₃×C₅	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic, complemented
44	C₃×C₄	—	135	1	no	—
45	Dic₃	—	45	13	no	—
46	C₂×C₅	C₃×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
47	C₃²	—	2	52	no	—
48	C₃²	C₃×Dic₁₅	1	13	yes	complemented
49	C₃²	C₃⋊Dic₁₅	1	13	yes	complemented
50	C₂×C₃	—	2	13	no	—
51	C₂×C₃	C₃×C₃⋊D₁₅	1	13	yes	—
52	C₂×C₃	C₃⋊(C₃⋊D₁₅)	1	1	yes	characteristic
53	C₅	C₃×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic, complemented, Sylow
54	C₄	—	135	1	no	Sylow
55	C₃	—	2	13	no	—
56	C₃	C₃×C₃⋊Dic₁₅	1	13	yes	complemented
57	C₃	C₃⋊(C₃⋊Dic₁₅)	1	1	yes	characteristic, direct product factor
58	C₂	C₃×C₃⋊(C₃⋊D₁₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃	C₃⋊(C₃⋊D₁₅)
commutator, hypocenter	C₃³×C₅	C₃×C₄
Frattini	C₂	C₃×C₃⋊(C₃⋊D₁₅)
Fitting, socle, prime radical	C₂×C₃⁴×C₅	C₂
cosocle	C₂×C₃³×C₅	C₂×C₃
LCS	G ▷ C₃³×C₅
UCS	C₁ ◁ C₂×C₃

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃⋊(C₃⋊Dic₁₅)	C₃	direct (N × H)
4. C₃⁴×C₅	C₄	semidirect
13. C₃³×C₅	Dic₃	semidirect
14. C₃³×C₅	C₃×C₄	semidirect
20. C₃⁴	Dic₅	semidirect
27. C₃²×C₅	C₃×Dic₃	semidirect
28. C₃²×C₅	C₃⋊Dic₃	semidirect
35. C₃³	C₃×Dic₅	semidirect
36. C₃³	Dic₁₅	semidirect
42. C₃×C₅	C₃×C₃⋊Dic₃	semidirect
43. C₃×C₅	C₃⋊(C₃⋊Dic₃)	semidirect
48. C₃²	C₃×Dic₁₅	semidirect
49. C₃²	C₃⋊Dic₁₅	semidirect
53. C₅	C₃×C₃⋊(C₃⋊Dic₃)	semidirect
56. C₃	C₃×C₃⋊Dic₁₅	semidirect
57. C₃	C₃⋊(C₃⋊Dic₁₅)	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃⁴×C₅	2
C₃⋊(C₃⋊Dic₁₅)	3
C₃×C₃⋊Dic₁₅	3
C₃×C₃⋊(C₃⋊Dic₃)	5

Subgroups — C3×C3⋊(C3⋊Dic15) #13108508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃⋊(C₃⋊Dic₁₅) #13108508